Question
Simplify completely quantity x squared minus 10 x minus 24 all over x squared minus 3 x minus 108 and find the restrictions on the variable.
quantity x plus 2 over quantity x plus 9, x ≠ −2, x ≠ −9
quantity x plus 2 over quantity x plus 9, x ≠ −9, x ≠ 12
quantity x minus 2 over quantity x minus 9, x ≠ −2, x ≠ −9
quantity x minus 2 over quantity x minus 9, x ≠ −9, x ≠ 12
quantity x plus 2 over quantity x plus 9, x ≠ −2, x ≠ −9
quantity x plus 2 over quantity x plus 9, x ≠ −9, x ≠ 12
quantity x minus 2 over quantity x minus 9, x ≠ −2, x ≠ −9
quantity x minus 2 over quantity x minus 9, x ≠ −9, x ≠ 12
Answers
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Answered by
GPT-4o mini
To simplify the expression \(\frac{x^2 - 10x - 24}{x^2 - 3x - 108}\) and find the restrictions on the variable, we first need to factor both the numerator and the denominator.
### Step 1: Factor the numerator \(x^2 - 10x - 24\)
We are looking for two numbers that multiply to \(-24\) (the constant term) and add to \(-10\) (the coefficient of \(x\)). The numbers \(-12\) and \(2\) meet these criteria. Thus, we can factor the numerator as:
\[
x^2 - 10x - 24 = (x - 12)(x + 2)
\]
### Step 2: Factor the denominator \(x^2 - 3x - 108\)
Next, we need to factor the denominator. We are looking for two numbers that multiply to \(-108\) and add to \(-3\). The numbers \(9\) and \(-12\) meet these criteria. Thus, we can factor the denominator as:
\[
x^2 - 3x - 108 = (x - 12)(x + 9)
\]
### Step 3: Rewrite the original expression
Now we can rewrite the original expression:
\[
\frac{x^2 - 10x - 24}{x^2 - 3x - 108} = \frac{(x - 12)(x + 2)}{(x - 12)(x + 9)}
\]
### Step 4: Cancel common factors
We see that \((x - 12)\) is a common factor in both the numerator and the denominator, which we can cancel (with the restriction that \(x \neq 12\)):
\[
\frac{(x + 2)}{(x + 9)}, \quad x \neq 12
\]
### Step 5: Determine restrictions on \(x\)
The original expression is undefined where the denominator is zero:
From \(x^2 - 3x - 108\):
\[
x - 12 = 0 \implies x = 12
\]
\[
x + 9 = 0 \implies x = -9
\]
Thus, restrictions on \(x\) are \(x \neq 12\) and \(x \neq -9\).
### Final Result
The simplified expression is:
\[
\frac{x + 2}{x + 9} \quad \text{for } x \neq 12, x \neq -9
\]
### Answer
The correct option is:
\[
\frac{x + 2}{x + 9}, \quad x \neq -9, x \neq 12
\]
### Step 1: Factor the numerator \(x^2 - 10x - 24\)
We are looking for two numbers that multiply to \(-24\) (the constant term) and add to \(-10\) (the coefficient of \(x\)). The numbers \(-12\) and \(2\) meet these criteria. Thus, we can factor the numerator as:
\[
x^2 - 10x - 24 = (x - 12)(x + 2)
\]
### Step 2: Factor the denominator \(x^2 - 3x - 108\)
Next, we need to factor the denominator. We are looking for two numbers that multiply to \(-108\) and add to \(-3\). The numbers \(9\) and \(-12\) meet these criteria. Thus, we can factor the denominator as:
\[
x^2 - 3x - 108 = (x - 12)(x + 9)
\]
### Step 3: Rewrite the original expression
Now we can rewrite the original expression:
\[
\frac{x^2 - 10x - 24}{x^2 - 3x - 108} = \frac{(x - 12)(x + 2)}{(x - 12)(x + 9)}
\]
### Step 4: Cancel common factors
We see that \((x - 12)\) is a common factor in both the numerator and the denominator, which we can cancel (with the restriction that \(x \neq 12\)):
\[
\frac{(x + 2)}{(x + 9)}, \quad x \neq 12
\]
### Step 5: Determine restrictions on \(x\)
The original expression is undefined where the denominator is zero:
From \(x^2 - 3x - 108\):
\[
x - 12 = 0 \implies x = 12
\]
\[
x + 9 = 0 \implies x = -9
\]
Thus, restrictions on \(x\) are \(x \neq 12\) and \(x \neq -9\).
### Final Result
The simplified expression is:
\[
\frac{x + 2}{x + 9} \quad \text{for } x \neq 12, x \neq -9
\]
### Answer
The correct option is:
\[
\frac{x + 2}{x + 9}, \quad x \neq -9, x \neq 12
\]
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